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Hierarchical clustering Agglomerative (Bottom-up) Compute all pair-wise pattern-pattern similarity coefficients Place each of n patterns into a class of its own Merge the two most similar clusters into one Replace the two clusters into the new cluster Re-compute inter-cluster similarity scores w.r.t. the new cluster Repeat the above step until there are k clusters left (k can be 1) Slide credit: Min Zhang

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Hierarchical clustering Divisive (Top-down) Start at the top with all patterns in one cluster The cluster is split using a flat clustering algorithm This procedure is applied recursively until each pattern is in its own singleton cluster

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Bottom-up vs. Top-down Which one is more complex? Which one is more efficient? Which one is more accurate?

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Bottom-up vs. Top-down Which one is more complex? Top-down Because a flat clustering is needed as a subroutine Which one is more efficient? Which one is more accurate?

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Bottom-up vs. Top-down Which one is more complex? Which one is more efficient? Which one is more accurate?

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Bottom-up vs. Top-down Which one is more complex? Which one is more efficient? Top-down For a fixed number of top levels, using an efficient flat algorithm like K-means, divisive algorithms are linear in the number of patterns and clusters Agglomerative algorithms are least quadratic Which one is more accurate?

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Bottom-up vs. Top-down Which one is more complex? Which one is more efficient? Which one is more accurate?

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Bottom-up vs. Top-down Which one is more complex? Which one is more efficient? Which one is more accurate? Top-down Bottom-up methods make clustering decisions based on local patterns without initially taking into account the global distribution. These early decisions cannot be undone. Top-down clustering benefits from complete information about the global distribution when making top-level partitioning decisions. Back

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Deciding K We can plot objective function values for k=1 to 6 The abrupt change at k=2 is highly suggestive of two clusters knee finding or elbow finding Note that the results are not always as clear cut as in this toy example Back Image: Henry Lin

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Spatially constrained KFCM : the set of neighbors that exist in a window around : the cardinality of controls the effect of the penalty term The penalty term is minimized when Membership value for x j is large and also large at neighboring pixels Vice versa 0.9 0.1 0.90.1 Equation: Dao-Qiang Zhang, Song-Can Chen

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Problem with min. cuts Minimum cut criteria favors cutting small sets of isolated nodes in the graph Not surprising since the cut increases with the number of edges going across the two partitioned parts Image: Jianbo Shi and Jitendra Malik

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Algorithm Given an image, set up a weighted graph and set the weight on the edge connecting two nodes to be a measure of the similarity between the two nodes Solve for the eigenvectors with the second smallest eigenvalue Use the second smallest eigenvector to bipartition the graph Decide if the current partition should be subdivided and recursively repartition the segmented parts if necessary